02 chapter 3
TRANSCRIPT
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Chapter 3
Full Vehicle Simulation Model
Two different versions of the full vehicle simulation model of the test vehicle
will now be described. The models are validated against experimental results.
A unique steering driver model is proposed and successfully implemented.
This driver model makes use of a non-linear gain, modelled with the Magic
Formula, traditionally used for the modelling of tyre characteristics.
3.1 Initial Vehicle Model
A Land Rover Defender 110 was initially modelled in ADAMS View 12
(MSC 2005) with standard suspension settings as a baseline. The ADAMS
521 interpolation tyre model is used, because of its ability to incorporate test
data in table format. The tyre’s vertical dynamics and load dependent lateral
dynamics are thus considered in this model. In order to keep the model
as simple as possible, yet as complex as necessary, longitudinal dynamic
behaviour of the tyres and vehicle is not considered here. The anti-roll
bar and bump stops are left unchanged. Only the spring and
damper characteristics are changed for optimisation purposes. This study
builds on current research into a two-state semi-active spring-damper system.
The semi-active unit has been included in the ADAMS model and replaces
the standard springs and dampers. The inertias of the vehicle body were
determined by scaling down data available for an armoured prototype Land
Rover 110 Wagon, and were considered to be representative of the lighter
vehicle.
23
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 24
The complete model consists of 16 unconstrained degrees of freedom, 23
moving parts, 11 spherical joints, 10 revolute joints, 9 Hooke’s joints, and
one motion defined by the steering driver. The vehicle direction of heading
is controlled by a simple single point steering driver, adjusting the steering
wheel rotation according to the difference of the desired course from the
current course at a preview distance ahead of the vehicle.
3.2 Refined Vehicle Model
A refined model of the Land Rover Defender 110 is also modelled
in MSC.ADAMS View (MSC 2005) with standard suspension settings,
as a baseline. For this model, the non-linear MSC.ADAMS Pacejka 89
tyre model (Bakker et al. 1989) is fitted to measured tyre data, and
used within the model. This tyre model was selected as it was found that
the 5.2.1 tyre model could not handle tyre slip angles larger than 3 degrees.
The Pacejka 89 tyre model was used with a point follower approximation for
rough terrain, to speed up the simulation speed, and as a result of limited
tyre and test track data available at the time. As in the initial model,
the tyre’s vertical dynamics and load dependent lateral dynamics are also
considered in this model. In order to keep the model as simple as possible,
yet as complex as necessary, longitudinal dynamic behaviour of the tyres
and vehicle is again not considered here. The vehicle body is modelled as
two rigid bodies connected along the roll axis at the chassis height, by a
revolute joint and a torsional spring, in order to better capture the vehicle
dynamics due to body torsion in roll. The anti-roll bar is modelled as a
torsional spring between the two rear trailing arms to be representative of the
actual anti-roll bar’s effect. The bump and rebound stops, are modelled with
non-linear splines, as force elements between the axles and vehicle body. The
suspension bushings are modelled as kinematic joints with torsional spring
characteristics that are representative of the actual vehicle’s suspension joint
characteristics, in an effort to speed up the solution time, and help decrease
numerical noise. The baseline vehicle’s springs and dampers are modelled
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 25
Table 3.1: MSC.ADAMS vehicle model’s degrees of freedom
Body Degrees of Freedom Associated Motions
Vehicle Body 7 body torsion
(2 rigid bodies) longitudinal, lateral, vertical
roll, pitch, yaw
Front Axle 2 roll, vertical
Rear Axle 2 roll, vertical
Wheels 4 x 1 rotation
with measured non-linear splines. The vehicle’s center of gravity (cg) height
and moments of inertia were measured (Uys et al. 2005) and used within
the model. Only the spring and damper characteristics are changed for
optimisation purposes. The 4S 4 unit has been included in the MSC.ADAMS
model, using the MSC.ADAMS Controls environment to include the Simulink
model, and replaces the standard springs and dampers. Due to the fact that
different suspension characteristics are being included the first two seconds
of the simulation have to be discarded, while the vehicle is settling into an
equilibrium condition. Figures 3.1 and 3.2 indicates the detailed kinematic
modelling of the rear and front suspensions. The complete model consists
of 15 unconstrained degrees of freedom, 16 moving parts, 6 spherical joints,
8 revolute joints, 7 Hooke’s joints, and one motion defined by the steering
driver. The degrees of freedom are indicated in Table 3.1.
The vehicle’s direction of heading is controlled by a carefully tuned yaw rate
steering driver, adjusting the front wheels’ steering angles according to the
difference of the desired course from the current course at a preview distance
ahead of the vehicle (see paragraph 3.4). The longitudinal driver is modelled
as a variable force attached to the body at wheel height depending on the
difference between the instantaneous speed and desired speed (see paragraph
3.3). This MSC.ADAMS model is linked to MATLAB (Mathworks 2000b)
through a Simulink block that requires as inputs the spring and damper
design variable values, and returns outputs of vertical accelerations, vehicle
body roll angle and roll velocity.
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 26
Legend
Hooke’s Joint
Revolute Joint
Spherical Joint
Rigid Body
Vehicle Front Body +6
Front Axle +6
Steering Link +6
S t e er i n g
A r m
+ 6
S t e er i n g
A r m
+ 6
Wh e el + 6
Wh e el + 6
L e a d i n gA r m
+ 6
P anh ar d r o d + 6
L e a d i n gA r m
+ 6
-4-3
-5
-4 -4 -4
-3 -3
-5
-5 -5
-4
-3
-5
Body +6
-4
Figure 3.1: Modelling of the full vehicle in MSC.ADAMS, front suspension
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 27
Legend
Hooke’s Joint
Revolute Joint
Spherical Joint
Rigid Body
Vehicle Rear Body +6
Rear Axle +6
Wheel +6 Wheel +6
T r ai l i n g
A r m
+ 6
-4 -4
-3 -3 -3
-5 -5
-4
-3
-5
Body +6
A - A r m
+ 6
T r ai l i n g
A r m
+ 6
-5
-5
Vehicle Front Body +6
Figure 3.2: Modelling of the full vehicle in MSC.ADAMS, rear suspension
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 28
3.2.1 Validation of Full Vehicle Model
The MSC.ADAMS full vehicle model is validated against measured test
results performed on the baseline vehicle. The measurement positions are
defined by Figure 3.3 and Table 3.2. The correlation results are presented
in Figure 3.4 for the baseline vehicle travelling over two discrete bumps to
evaluate vertical dynamics, and in Figure 3.5 for the vehicle performing a
double lane change manœuvre at 65 km/h. From the results it is evident
that the model returns excellent correlation to the actual vehicle. It is,
however, computationally expensive to solve and exhibits severe numerical
noise due to all the included non-linear effects.
Table 3.2: Land Rover 110 test points
channel point position measure axis
1 B center of gravity velocity longitudinal
2 G left front bumper acceleration longitudinal
3 lateral
4 vertical
5 C rear passenger acceleration longitudinal
6 lateral
7 vertical
8 I right front bumper acceleration vertical
9 A steering arm displacement relative arm/body
10 D left rear spring displacement relative body/axle
11 E right rear spring
12 F left front spring
13 H right front spring
14 B center of gravity angular velocity roll
15 pitch
16 yaw
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 29
Figure 3.3: Test vehicle indicating measurement positions
3.3 Vehicle Speed Control
The speed control is modelled as a variable force F drive attached to the body
at wheel center height. The magnitude of this force depends on the difference
between the instantaneous speed ẋact and desired speed ẋd. Because the
vehicle is a four-wheel drive with open differentials, the vertical tyre force
F ztyre is measured at all tyres (1 to 4). If a tyre looses contact with the
ground, the driving force to the vehicle is removed until all wheels are again
in contact with the ground. The driving force is thus defined as:
if F ztyre1→4 = 0
F drive = 0
else
F drive = 4min(1200,1200(ẋd−ẋact))
0.4
end
(3.1)
The gain value of 1200 was determined to be sufficiently large to ensure
fast stable acceleration of the vehicle model from rest up to the desired
simulation speed. This force is multiplied by 4 as there is one force acting on
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 30
2 4 6 8 10−40
−20
0
20
40
time [s]
p i t c h v e
l o c i t y [ o / s ]
2 4 6 8 10−100
−50
0
50
100
time [s]
l r s p r i n g d i s p
l a c e m e n t [ m m ]
2 4 6 8 10
−50
0
50
100
time [s]
r f s p r i n g d i s p l a c
e m e n t [ m m ] Vehicle Test
ADAMS Model
2 4 6 8 10−1
−0.5
0
0.5
1
1.5
time [s]
l r v e r t i c a l a c c e
l e r a t i o n [ g ]
Figure 3.4: Discrete bumps, 15 km/h, validation of MSC.ADAMS model’s
vertical dynamics
the vehicle representative of the torque applied to the four wheels, and 0.4
meters is the radius of the tyres. The MSC.ADAMS model is then linked to
the Simulink (Mathworks 2000b) based driver model that returns as outputs
the desired vehicle speed and steering angle, calculated using the vehicle’s
dynamic response.
3.4 Driver Model For Steering Control
The use of driver models for the simulation of closed loop vehicle handling
manœuvres is vital. However, great difficulty is often experienced
in determining the gain parameters for a stable driver at all speeds, and
vehicle parameters. A stable driver model is of critical importance during
mathematical optimisation of vehicle spring and damper characteristics
for handling, especially when suspension parameters are allowed to
change over a wide range. The determination of these gain factors becomes
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excellent correlation to test results.
The primary reason for requiring a driver model in the present study, is
for the optimisation of the vehicle’s suspension system. The suspension
characteristics are to be optimised for handling, while performing the closed
loop ISO3888-1 (1999) double lane change manœuvre. The driver model
thus has to be robust for various suspension setups, and perform only one
simulation to return the objective function value. Thus steering controllers
with learning capability will not be considered, as the suspension could
be vastly different from one simulation to the next. Only lateral path
following is considered in this preliminary research, as the double lane change
manœuvre is normally performed at a constant vehicle speed.
Previous research into lateral vehicle model drivers, was conducted amongst
others by Sharp et al. (2000) who implement a linear, multiple preview
point controller, with steering saturation limits mimicking tyre saturation,
for vehicle tracking. The vehicle model used is a 5-degree-of-freedom (dof)
model, with non-linear Magic Formula tyre characteristics, but no suspension
deflection. This model is successfully applied to a Formula One vehicle
performing a lane change manœuvre. Also Gordon et al. (2002) make use of
a novel method, based on convergent vector fields, to control the vehicle along
desired routes. The vehicle model is a 3-dof vehicle, with non-linear Magic
Formula tyre characteristics, but with no suspension deflection included.
The driver model is successfully applied to lane change manœuvres.
The primary similarity between these methods is that vehicle models with no
suspension deflection were used. The current research is, however, concerned
with the development of a controllable suspension system for Sports Utility
Vehicles (SUV’s). The suspension system thus has to be modelled, and the
handling dynamics simulated for widely varying suspension settings. The
vehicle in question has a comparatively soft suspension, coupled to a high
center of gravity, resulting in large suspension deflections when performing
the double lane change manœuvre. This large suspension deflection, results
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 33
in highly unstable vehicle behaviour, eliminating the use of driver models
suited to vehicles with minimal suspension deflection. Steady state rollover
calculations also show that the vehicle will roll over before it will slide.
Proköp (2001) implements a PID (Proportional Integral Derivative)
prediction model for tracking control of a bicycle model vehicle. The driver
model makes use of a driver plant model that is representative of the actual
vehicle. The driver plant increases in complexity to perform the required
dynamic manœuvre, from a point mass to a four wheel model with
elastokinematic suspension. This model is then optimised with the SQP
(Sequential Quadratic Programming) optimisation algorithm for each time
step. This approach, however, becomes computationally expensive, when
optimisation of the vehicle’s handling is to be considered.
For the current research several driver model approaches were implemented,
but with limited success. Due to the difficulty encountered with
the implementation of a driver model for steering control, it was decided
to characterize the whole vehicle system, using step steer, and ramp steer
inputs, and observe various vehicle parameters. This lead to the discovery
that the relationship between vehicle yaw acceleration vs. steering rate for
various vehicle speeds appeared very similar to the side force vs. slip angle
characteristics of the tyres. With this discovery it was decided to implement
the proposed novel driver model, with the non-linear gain factor modelled
with the Pacejka Magic Formula, normally used for tyre data.
3.4.1 Driver Model Description
To investigate the relationship between vehicle response and steering inputs,
simulations were performed for various steering input rates (Figure 3.6, where
ts is the start of the ramp when the vehicle has reached the desired speed), at
various vehicle speeds. It was found that there existed a trend very similar
to the tyre’s lateral force vs. slip angle at various vertical loads, (Figure
3.7) with the vehicle’s yaw acceleration vs. steering rate at different vehicle
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 34
speeds (Figure 3.8). Because of this relationship it was postulated that the
vehicle could be controlled by comparing the actual yaw acceleration to the
desired yaw acceleration, and adjusting the steering input rate.
tts
δ
δ
Figure 3.6: Vehicle characterisation steering input
From dynamics principles it is known that, for a rigid body undergoing
motion in a plane, the rotational angle as a function of time is dependant
on: the current rotational angle ϑ0, the current rotational velocity ϑ̇, the
rotational acceleration ϑ̈, and the time step δt over which the rotational
acceleration is assumed constant. If the rotational acceleration is not constant,
but sufficiently small time steps are considered, the predicted rotational angle
ϑ p will be sufficiently well approximated. The predicted rotational angle can
be determined as follows:
ϑ p = ϑ0 + ϑ̇δt + 1
2ϑ̈δt2 (3.2)
The above equation can be modified for a vehicle’s yaw rotation motion by
defining ϑ as the yaw angle ψ. Considering Figure 3.9, the driver model
parameters can now be defined as:
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0 2 4 6 8 10 12 140
2000
4000
6000
8000
10000
12000
slip angle [o]
l a t e r a l f o r c e [ N ]
Fz 1 kNFz 10 kNFz 20 kNFz 30 kNFz 40 kNvertical load
Figure 3.7: Tyre’s lateral force vs. slip angle characteristics for different vertical
loads
0 1 2 3 4 5 6 7 8 90
5
10
15
20
25
30
35
40
45
50
steer rate [o /s]
y a w
a c c e l e r a t i o n [ o / s 2 ]
Vehicle Response
10
30507090speed[km/h]
Figure 3.8: Vehicle yaw acceleration response to different steering rate inputs
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 36
• desired yaw angle of the vehicle ψd, equivalent to ϑ p
• actual vehicle yaw angle ψa, equivalent to ϑ0
• actual vehicle yaw rate ψ̇a, equivalent to ϑ̇
• response/preview time τ , equivalent to δt
• vehicle forward velocity ẋ
x
aψ
d ψ
aψ
previewd xτ =
x
y
desired path
vehicle
Figure 3.9: Definition of driver model parameters
The yaw acceleration ψ̈ needed to obtain the desired yaw angle is calculated
from equation (3.2), substituting in the equivalent variables, as follows:
ψ̈ = 2ψd − ψa − ψ̇aτ
τ 2 (3.3)
The vehicle’s steady state yaw acceleration ψ̈ with respect to different steering
rates δ̇ , was determined for a number of constant vehicle speeds ẋ and is
presented in Figure 3.8. Where the vehicle’s response did not reach
steady-state, and the vehicle slided out, or rolled over, the yaw acceleration
just prior to loss of control was used. This process is computationally
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 37
expensive as 11 different steering ramp rates, for each vehicle speed, were
applied to the vehicle model and simulated. The steady state yaw acceleration
reached was then used to generate the figure. When comparing Figure
3.8 to the vehicle’s lateral tyre characteristics, presented in Figure 3.7, it
appears reasonable that the Magic Formula could also be fitted to the steering
response data. Therefore the reformulated Magic Formula, discussed below,
is fitted to this data, and returns the required steering rate δ̇ , which is defined
as:
δ̇ = f (ψ̈, ẋ) (3.4)
As output, the driver model provides the required steering rate δ̇ , which is
then integrated for the time step δt to give the required steering angle δ .
The Magic Formula is fitted through the obtained data, as it is a continuous
function over the fitted range. Normal polynomial curve fits would be discreet
for the vehicle speed they are fitted to and an interpolation scheme would
be necessary for in-between vehicle speeds. The Magic Formula is thus a
continuous approximation described by 12 values, as opposed to multiple
curve formulae, requiring intermediate interpolation.
3.4.2 Magic Formula Fits
The Magic Formula was proposed by Bakker et al. (1989) to describe the
tyre’s handling characteristics in one formula. In the current study the
Magic Formula will be considered in terms of the tyre’s lateral force vs. slip
angle relationship, which directly affects the vehicle’s handling and steering
response. The Magic Formula is defined as:
y(x) = Dsin(Carctan{Bx − E (Bx − arctan(Bx))})
Y (X ) = y(x) + S v
x = X + S h
(3.5)
The terms are defined as:
• Y (X ) tyre lateral force F y
• X tyre slip angle α
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 38
• B stiffness factor
• C shape factor
• D peak factor
• E curvature factor
• S h horizontal shift
• S v vertical shift
These terms are dependent on the vertical tyre load F z and camber angle
γ . The lateral force F y vs. tyre slip angle α relationship typically takes onthe shape as indicated in Figure 3.7, for different vertical loads. Considering
the shape of Figure 3.8 presenting the yaw acceleration vs. steering rate for
different vehicle speeds, the Magic Formula can be successfully fitted, with
the parameters redefined as:
• vertical tyre load F z is equivalent to vehicle speed ẋ
• tyre slip angle α is equivalent to steering rate δ̇
• tyre lateral force F y is equivalent to vehicle yaw acceleration ψ̈
The Magic Formula for the vehicle’s steering response can now be stated as:
y(x) = Dsin(Carctan{Bx − E (Bx − arctan(Bx))})
Y (X ) = y(x) + S v
x = X + S h
(3.6)
With the terms defined as:
• Y (X ) yaw acceleration ψ̈
• X steering rate δ̇
• B stiffness factor
• C shape factor
• D peak factor
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 39
• E curvature factor
• S h horizontal shift
• S v vertical shift
With the redefined parameters, the Magic Formula coefficients can
be determined in the usual manner. The determination of the coefficients
applied for the steering driver is now discussed. The baseline vehicle’s response
as indicated in Figure 3.8 is used for the fitting of the parameters.
3.4.3 Determination of Factors
The peak factor D is determined by plotting the maximum yaw acceleration
value ψ̈ against the vehicle speed ẋ. For this the graphs have to be interpolated.
Quadratic curves were fitted through the vehicle’s response curves, and the
estimated peak values were used. The peak factor is defined as:
D = a1 ẋ2 + a2 ẋ (3.7)
The peak factor curve was fitted through the estimated peak values, with
emphasis on accurately capturing the data for vehicle speeds of 50 to 90
km/h. The 90 km/h peak was taken as the point where the graph changed
due to the maximum yaw acceleration just prior to roll-over. The resulting
quadratic curve fit to the predicted peak values of the yaw acceleration is
shown in Figure 3.10. It is observed that the fit for the Magic Formula is
poor for 30 km/h. This is attributed to the almost linear curve fit through
the yaw acceleration vs. steering rate for speeds of 10 and 30 km/h Figure
3.6, resulting in an unrealistically high prediction of the peak values.
In the original paper (Bakker et al. 1989), BC D is defined as the cornering
stiffness, here it will be termed the ‘yaw acceleration gain’. For the yaw
acceleration gain the gradient at zero steering rate is plotted against vehicle
speed as illustrated in Figure 3.11. The camber term γ of the original paper
will be ignored so that coefficient a5 becomes zero. The yaw acceleration
gain is fitted with the following function:
BC D = a3sin(2arctan(ẋ/a4))(1 − a5γ ) (3.8)
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 40
10 20 30 40 50 60 70 80 900
50
100
150
200
250
speed [km/h]
p e a k y a w
a c c l [ ( o / s 2 ) ]
actualMF fit
Figure 3.10: Magic Formula coefficient quadratic fit through equivalent peak
values
For the determination of the curvature E , quadratic curves were fitted through
each of the curves in Figure 3.8. These approximations could then be
differentiated twice to obtain the curvature for each. This curvature is plotted
against vehicle speed ẋ, and the straight line approximation:
E = a6 ẋ + a7 (3.9)
is then fitted through the data points, in order to determine the coefficient a6
and a7. The straight line approximation fitted through the points is shown
in Figure 3.12.
The shape factor C , is determined by optimising the resulting Magic Formula
fits to the measured data. This parameter is the only parameter that has
to be adjusted in order to achieve better Magic formula fits to the original
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 41
10 20 30 40 50 60 70 80 900
2
4
6
8
10
12
14
16
speed [km/h]
y a w
a c c e l e r a t i o n g a i n [ ( o / s 2 ) / ( o / s ) ]
actualMF fit
Figure 3.11: Magic Formula fit of yaw acceleration gain through the actual data
10 20 30 40 50 60 70 80 90−2
−1.5
−1
−0.5
0
0.5
speed [km/h]
c u r v a t u r e [ ( o / s 2 ) / ( o 2 / s 2 ) ]
actualMF fit
Figure 3.12: Determination of curvature coefficients
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 42
data. It is defined in terms of the Magic Formula coefficient a0 as follows:
C = a0 (3.10)
The stiffness factor B is determined by dividing BC D by C and D:
B = BCD/CD (3.11)
In the current research the horizontal and vertical shift of the curves were
ignored allowing coefficients a8 to a13 to be assumed zero. The Magic Formula
fits to the original data are presented in Figure 3.13. It can be seen that
most of the fits except for 90 km/h are very good. The vehicle simulation
failed for most of the steering rate inputs before reaching a steady state yaw
acceleration at 90 km/h, thus this can be viewed as an unstable regime.
With the Magic formula coefficients being determined, the manipulation of
the Magic formula for the driver application is discussed.
0 2 4 6 8 10 120
5
10
15
20
25
30
35
40
45
50
55
steer rate [o /s]
y a w
a c c e l e r a t i o n [ o / s 2 ]
10 act10 fit30 act30 fit50 act50 fit70 act70 fit90 act
90 fitspeed[km/h]
Figure 3.13: Magic Formula fits to original vehicle steering behaviour
3.4.4 Reformulated Magic Formula
The steering driver requires, as output, the steering rate δ̇ . For this reason
the Magic Formula’s subject of the formula must be reformulated, to make
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 44
2 4 6 8 10−40
−20
0
20
40
time [s]
l r s p r i n g d i s p l a c e m e n t [ m m ]
2 4 6 8 10−30
−20
−10
0
10
20
30
time [s]
y a w
v e l o c i t y [ o / s ]
2 4 6 8 10−1
−0.5
0
0.5
1
time [s]
l f l a t e r a l
a c c e l e r a t i o n [ g ]
2 4 6 8 10−4
−2
0
2
4
time [s]
k i n g p i n a
n g l e [ o ]
Vehicle TestMF Driver
Figure 3.14: Correlation of Magic Formula driver model to vehicle test at an
entry speed of 63 km/h
The driver model was then analysed for changing the vehicle’s suspension
system from stiff to soft, for various speeds. Presented in Figure 3.15 is the
driver model’s ability to keep the vehicle at the desired yaw angle (Genta
1997) over time. From the results it can be seen that a varying preview
time with vehicle speed, would be beneficial, however, it is felt that for this
preliminary research the constant 0.5 seconds preview time is sufficient. Also
it is evident that the softer suspension system, and 70 km/h vehicle speed,
are slightly unstable, as seen by the oscillatory nature at the end of the
double lane change manœuvre.
The results show that the driver model provides a well controlled steering
input. Also there is a lack of high frequency oscillation typically associated
with single point preview driver models, when applied to highly non-linear
vehicle models like SUV’s, that are being operated close to their limits in the
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CHAPTER 3. FULL VEHICLE SIMULATION MODEL 45
double lane change manœuvre.
3.5 Conclusions
It has been shown that the Magic Formula, traditionally used for describing
tyre characteristics, can be fitted to the vehicle’s steering response, in the
form of yaw acceleration vs. steering rate, for different vehicle speeds.
A single point steering driver model has been successfully implemented on a
highly non-linear vehicle model. The success of the driver model, is attributed
to the modelling of the vehicle’s response with the Magic Formula. The
success of the single point steering driver can be related to the non-linear gain
factor, that changes in value with vehicle speed and required yaw acceleration.
Future work should entail an investigation into determining the parameters of
the vehicle that modify the tyre characteristic Magic Formula coefficients to
arrive at the steering rate and yaw acceleration parameters. Ideally the tyre
Magic Formula coefficients should be multiplied by some modifying factor,
based on vehicle characteristics, to be used directly for the control of the
vehicle steering. This would eliminate the need for the computationally
expensive characterisation currently required. A further aspect that could
be considered is determining the value of varying preview time with vehicle
speed. The driver model is, however, sufficiently robust to be used in the
optimisation of the vehicle’s suspension characteristics for handling.
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60 80 100 120 140 160 180−12
−10
−8
−6
−4
−2
0
2
4
6
8
10
12
14
x dist [m]
y a w
a n g l e [ o ]
stiff 50 km/hmid 60 km/hsoft 40 km/hstiff 70 km/h
desired
Figure 3.15: Comparison of different suspension settings and vehicle speeds, for
the double lane change manœuvre, where the desired is as proposed
by Genta (1997)